As @imjolly has already said, if you already have a master socket an engineer won't visit. Any internal cabling from the master socket you will have to do yourself unless you have paid extra for an engineer install.
DSL connectivity is RJ11 regardless of what device you are using. Modem to router is RJ45.
It is always better to have as short a RJ11 connection as possible.
Be interesting to know if you actually do get engineer visit and a socket rewire
We had issues with where our BT socket was and initially used a long RJ11 cable to a BT Hub but was unstable even with a quality cable. In the end i went for a Draytek Vigor Modem then used a long Cat7 cable from there to the Router TP-Link TL-R600VPN. Our set up is in the Cellar and so could go directly though the floor to the Modem, since i have had this set up (There is more but not relavent to this thread) its been extreamly stable so the key i found is to have the shortest RJ11 cable from modem to BT house socket and never to use extenstions or long lengths of RJ11.
Hope this helps
You mentioned in post 13 that you had an online chat with someone at BT, did you keep the chat log?
Assuming the Engineer does come out the best possible scenario is that they will move your existing socket to a better location using CW1308 at no charge to yourself. I would suggest that IF the engineer does arrive to politely expain the socket location being an issue, and ask if he/she can move it for you.
The reason for asking about the log was that you have a record of the visit being Free of Charge.
Taking into account Message 17 and that BT will move your socket for free that could be an ideal solution, other than that you would have to supply your own Cat cable to transfer from modem to seperate router in the house.